Nilpotent Infinitesimals and Synthetic Differential Geometry in Classical Logic

Abstract

An extension of ℝ (and *ℝ) with nilpotent infinitesimals (e.g. h ≠ 0 but h ² = 0) is presented in order to obtain results similar to KockLawvere’s synthetic differential geometry [3], but in a classical and not intuitionistic context. The same extension can be used to add new infinitesimal points to spaces similar to Chen’s ones [1]. In the category of extended spaces we can develop differential geometry not only of usual manifolds but also of infinite dimensional spaces, without coordinates and with a strong geometric intuition, that is in a way that we will call “synthetic”.